The Illusion of Mathematical Certainty

Nate Silver’s questionable foray into predicting World Cup results got me thinking about the limitations of maths in economics (and the social sciences in general). I generally stay out of this discussion because it’s completely overdone, but I’d like to rebut a popular defence of mathematics in economics that I don’t often see challenged. It goes something like this:

Everyone has assumptions implicit in the way they view the world. Mathematics allows economists to state our assumptions clearly and make sure our conclusions follow from our premises so we can avoid fuzzy thinking.

I do not believe this argument stands on its own terms. A fuzzy concept does not become any less fuzzy when you attach an algebraic label to it and stick it into an equation with other fuzzy concepts to which you’ve attached algebraic labels (a commenter on Noah Smith’s blog provided a great example of this by mathematising Freud’s Oedipus complex and pointing out it was still nonsense). Similarly, absurd assumptions do not become any less absurd when they are stated clearly and transparently, and especially not when any actual criticism of these assumptions is brushed off the grounds that “all models are simplifications“.

Furthermore, I’m not convinced that using mathematics actually brings implicit assumptions out into the open. I can’t count the amount of times that I’ve seen people invoke demand-supply without understanding that it is built on the assumption of perfect competition (and refusing to acknowledge this point when challenged). The social world is inescapably complex, so there are an overwhelming variety of assumptions built into any type of model, theory or argument that tries to understand it. These assumptions generally remain unstated until somebody who is thinking about an issue – with or without mathematics – comes along and points out their importance.

For example, consider Michael Sandel’s point that economic theory assumes the value or characteristics of commodities are independent of their price and sale, and once you realise this is unrealistic (for example with sex), you come to different conclusions about markets. Or Robert Prasch’s point that economic theory assumes there is a price at which all commodities will be preferred to one another, which implies that at some price you’d substitute beer for your dying sister’s healthcare*. Or William Lazonick’s point that economic theory presumes labour productivity to be innate and transferable, whereas many organisations these days benefit from moulding their employees’ skills to be organisation specific. I could go on, but the point is that economic theory remains full of implicit assumptions. Understanding and modifying these is a neverending battle that mathematics does not come close to solving.

Let me stress that I am not arguing against the use of mathematics; I’m arguing against using gratuitous, bad mathematics as a substitute for interesting and relevant thinking. If we wish to use mathematics properly, it is not enough to express properties algebraically; we have to define the units in which these properties are measured. No matter how logical mathematics makes your theory appear, if the properties of key parameters are poorly defined, they will not balance mathematically and the theory will be logical nonsense. Furthermore, it has to be demonstrated that the maths is used to come to new, falsifiable conclusions, rather than rationalising things we already know. Finally, it should never be presumed that stating a theory mathematically somehow guards that theory against fuzzy thinking, poor logic or unstated assumptions. There is no reason to believe it is a priori desirable to use mathematics to state a theory or explore an issue, as some economistsseem to think.

*This has a name in economics: the axiom of gross substitution. However, it often goes unstated or at least underexplored: for example, thesetwo popular microeconomics texts do not mention it all.

The rate of profit in perfect competition is not actually zero: there is still a return to capital, as there is a return to labour. Nevertheless, it is true that even a moderate level of profit in neoclassical economics is considered an inefficient rent, contrary to how some libertarians seem to use it.

As allready Gunnar Myrdal’s pointed out long ago in his stance in favour of economics as an intergrated social theory:

“I have no illusions that it will ever be possible to fit a
general theory into a neat econometric model. The relevant variables and the relevant relations between them are too many to permit that sort of heroic
simplification. This does not mean, however, that particular problems could not with advantage be treated in this way – provided that the variables and assumptions were selected on the basis of such insight into essential facts and
relations as only a general theory can furnish”
Gunnar Myrdal, (1957), Economic Theory and Underdeveloped Regions, London: University Paperbacks, Methuen.

Allready Pareto paved the way for this attempt. Later on, according to Myrdal there was a proliferation of terminological
and mathematical innovation, which presented general equilibrium and welfare theories in a new form, that is through a new analytical language. In spite of this kind of escapism, however, this body of neoclassical economic thought still
incorporates as he wrote:

“One version or another of the old, discredited rationalistic psychology and utilitarian moral philosophy. By implying them – as practical conclusions make evident that it does – it becomes unfounded and false”
Gunnar Myrdal. (1970), Objectivity in Social Research, London: Duckworth

In perfect competition, the interest rate will be equal to the return to capital, yes, otherwise there can be gains made by investing more in one or the other.

This implies zero profit if you define profit as ‘anything above the return to capital’, as is the case in economics. However, the return to capital still potentially covers risk, costs, entrepreneurial labour etc.

It’s not concepts that cannot be fuzzy. That is one thing you cannot escape, for you are dealing, in the case of utility, with a reality which is not directly observable, at the moment. However, the relationships, what we really want to understand, cannot be derived from fuzzy logic. How you come to conclusions has to be laid bare for everyone to examine. That is one thing that it’s hard to achieve satisfactorily without mathematical language.

Also, people often don’t mention assumptions when they are trying to explain economics to laymen, for obvious reasons, or speaking about issues casually. In classes (mine, at least), they are mentioned, albeit not thoroughly explored. Still, if one understands the mathematical concepts one is using, most of them are evident.

It’s not concepts that cannot be fuzzy. That is one thing you cannot escape, for you are dealing, in the case of utility, with a reality which is not directly observable, at the moment.

The fact that we are dealing with realities that are not directly observable is exactly my problem! Even if we derive a precise mathematical relationship between something like utility and outcomes, I don’t see how it’s logical when utility is a largely imaginary construct whose units bear no relation to anything else in the model. No matter how sophisticated the maths, using incommensurable parameters is like saying if you add 3 fish to 20 kilograms you get a shovel.

Still, if one understands the mathematical concepts one is using, most of them are evident.

This is not the same as stating assumptions clearly and explicitly, though. You’re saying that competent students/economists should be able to infer assumptions from the models, but this is little (if at all better) than having to infer assumptions from written theories. See my 3 examples .

“Logic” is more than mathematical models that make assumptions that are often naive. Decision making by top executives must often be made post haste. To be mostly successful decisions requires intuition that only exceptional minds possess. The ‘rational man’ premise that is taught in most MBA programs probably works well at lower levels of managers but creativity often just “happens” intuitively and this capability is available to very few managers and top executives. Mathematical economists should never be ‘at the table’ when determining decisions must be made quickly. Very few economic theories are practical enough to guide important decisions makers. At the macro governmental policy level, public officials seldom are educated enough in macroeconomics to make the right decisions at the appropriate time given measurable conditions. Finally, all economics majors or Ph.Ds assume that they are expert advisors but reality shows that swagger to be mostly incorrect. Go to a library and pick some academic articles in economics, read a couple, then ask what this article can do to advise important decisons. I guarantee your will leave the library shaking your head wondering whether these “economists” all came from some foreign planet where they all learned the economotalk so that they converse only with each other. I came from that planet, earned a Ph.D. in economics, and then started my first new business; then in a long lifetime founded three successful others. I also learned how to teach both economics and business subject matter so that both solid theory and practical implementation can be understood and applied. Even econometrics can be understood if taught so that it too can be understood for its theoretical but also practical uses. Keep the naive theoretical-only economists in their closets and wish them happy and contented lives.

Good comment. The question of what exactly economics is good for is a valid one. As you point out, it’s too convoluted and complex to be of use in most everyday decision making. It doesn’t seem to have covered itself in glory with advising policy. And its ability to forecast – whether conditional forecasts or precise forecasts of GDP/inflation etc. – is famously a joke. At best, it only seems to have extremely narrow applications, such as designing auction markets or determining whether some training program for the unemployed was successful. This isn’t nothing, but it’s far less than economics sets itself up as able to achieve.

“I’m not convinced that using mathematics actually brings implicit assumptions out into the open.”

I’ve seen ample evidence of economists using math as a trump card to circumvent reflection on their implicit assumptions. The “labor-leisure choice model” is a wonderful example of a Trojan horse loaded with counter-theoretical “simplifying” assumptions that economists are blythely unaware of. How could they be? They don’t know what the theory is, so how could they possibly be aware that the model they are incorporating implicitly assumes a special condition that “provisionally” suspends the theory because it is not mathematically tractable? That is, they mistake the model to be a faithful representation, not of reality, but of theory, which it blatantly is not.

People complain that economic models are unrealistic. It’s worse than that. They not even based on theory but on conventions that have been substituted for theory. The math keeps that all behind a screen.

They don’t know what the theory is, so how could they possibly be aware that the model they are incorporating implicitly assumes a special condition that “provisionally” suspends the theory because it is not mathematically tractable? That is, they mistake the model to be a faithful representation, not of reality, but of theory, which it blatantly is not.

I don’t follow which assumption you’re referring to – could you please elaborate?

A handful of good economists know this. But unfortunately in economics you really are rewarded for bamboozling by mathematics, even if the the concepts being translated into variables/parameters are known to be bogus, and the situation abstracts away from known real-world constraints.

So you end up with a large section of the profession wasting their careers arguing about the internal consistency of models most people know are wrong, or irrelevant, from the starting premise/assumptions used.

You are certainly right that maths is used to disguise terrible assumptions (or what I would call the translation of real life concepts into maths). Because EVERY model is only a collection of assumptions. Nothing more. Solving the model simply restates the assumptions in a different way.

Unless you can verify that the assumptions represent observable concepts, you are merely assuming a solution.

In fact, some pragmatic economists doing empirical and experimental work knowingly exceed the limits of their approach by reverse engineering their results into some kind utility function. You can interpret this as signalling to their audience more so than advancing knowledge.

Here’s the thought process.

“I’ve got an interesting result here. But look, I can squeeze it into a utility function. Don’t worry, I’m not using this result to challenge the premise of utility maximisation. I’m here to reinforce your belief in it, but expand your ability to apply this belief to new and unusual situations”

My personal view is that this is bad science. Well-trained social scientists should be aware of the limitations of their methods.

Just like stating the Oedipus thing as an equation does not make it less nonsensical, to leave it in its original literary form does not correct its lack of sense. Either way, it’s nonsense; maths does not add or subtract to it (pun intended :-).

Based on that, I’d say there is little, if any, connection between the use of mathematics and nonsense.

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Being punctilious with Paul Krugman, in “Two Cheers for Formalism”, which you linked to at the end of your post, he cites Deirdre McCloskey as a critic of “the excessive formalism of economics”. He cites McCloskey’s 1997 book “The Vices of Economics: The Virtues of the Bourgeoisie”.

Maybe McCloskey’s views changed between 1997 and 2002, but in 2002 she published a booklet called “The Secret Sins of Economics” where she identifies three virtues misidentified as sins. One of these virtues is the use of mathematics (quantification is another).

She explains the role of mathematics in economics and contrasts it to the use of quantification:

“Statistics or other quantitative methods in science (such as accounting or experiment or simulation) answer inductively How Much. Mathematics by contrast answers deductively Why, and in a refined and philosophical version very popular among mathematicians since the early nineteenth century, Whether.”

I have little to disagree with that, but I’ll just put it more plainly: in economics, mathematics answer qualitative questions.

To me, this just seems like an abuse of mathematics as tool, a way to pretend that economists have achieved something they have not by cloaking it in a language that is prima facie more ‘difficult’ to understand. Mathematics can answer qualitative questions, certainly, but it cannot answer nonsensical questions! And it should not be used to answer any type of question, because not all questions are amenable to mathematics.